On Weakly Noetherian N-Groups and Ascending Chain Conditions on Essential Ideals

Abstract

Several researchers like Armendariz, Efraim, Dunget al, Page, S.S, Yousif, Fuller, Anderson have been studied chain conditions on essential ideals in modules. We extend the concept of ascending chain condition to Near-ring groups defining weakly Noetherian N-group. Introducing almost weakly Noetherian N-group we established that for dgnr N, N-group E is almost weakly Noetherian, E has A.C.C. on essential ideals and ⁄ is weakly Noetherian for every essential ideal M of E are equivalent. These relations motivate us to study the relation between weakly Noetherian N-group E and A.C.C. on essential ideals of E. Finally we have shown Near-ring N is weakly Noetherian if iIEi of injective N-groups is injective.